10 REPRESENTATIONS OF ALGEBRAS OF CONTINUOUS FUNCTIONS 387
(with respect to the measure /z). For it follows from (13.12.2) that
l|A^(w)ll ^N00(w); also, for every positive real number a < NJw), there
exists a nonnegligible integrable subset P of K such that \u(t) \ ^ a for all
t e P; hence N2(w<pp) £ a • N2(<pP), from which (15.10.2) follows.
The measure \JL is not uniquely determined. We shall come back to this
point a little later (15.10.7).
The definition of MM(w) given in (15.10.1), and the formula (15.10.2), still
make sense if u is not necessarily continuous on K, but simply ^-measurable
and bounded in measure (by virtue of (13.12.5)). The mapping MH*Af (i/)
extended in this manner is a representation of the algebra with involution
«5?c(K> p) on the Hilbert space Lc(K, JH). Clearly, if u^ and u2 are equal
almost everywhere with respect to /*, we have M^u^ = M^(u2). Usually we
shall restrict the representation MM to a self-adjoint subalgebra of &% (K, u)
which does not depend on the choice of /x, namely the algebra *C(K) of
universally measurable, bounded complex-valued functions on K (13.9). By
(13.9.8.1) this is indeed a C-algebra with involution, and it is a Banach
subalgebra of J^(K) by virtue of Egoroff's theorem (13.9.10).
(15.10.3) Again let u^T(u) be a representation of ^(K) in a separable
Hilbert space E, admitting a totalizing vector a. (These hypotheses will be in
force up to and including (15.10.7).) The preceding remarks show that the
representation T may be extended to the algebra with involution ^C(K). The
extended representation (also denoted by T) does not depend on the choice
of totalizing vector a.
To prove this assertion, let x and y be two vectors belonging to E. For
each we^c(K), we have ||T(u)|| £ \\u\\ (15.5.7) and therefore
(15.10.3.1) \(T(u)-x\y)\^ \\u\\ -\\x\\- \\y\\;
consequently the linear form wh~>(T(w) • x \ y) is a measure uXiy on K such that
il/Oi ^ W'WI-
From this definition it follows immediately that
(15.10.3.2) ft.* = £,.,-
Also, if x = T(v) • a and y = T(w) * a, where y and w are in ^C(K), tnen f°r
each function u e ^C(K) we have
(T(u) • x\y) = (T(ifl?) • fl j T(w) • a) = (T(wuv) • 0| a) =
daement e. Let P be the set of positive linear forms on A (or of traces on A), which is a